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Equivalence Relations

Recall from the Partial Orders and Strict Partial Orders on Sets page that if $X$ is a set then a relation $R \subseteq X \times X$ is called a partial order on $X$ if $R$ is reflexive, antisymmetric, and transitive. We also defined $R$ to instead be a strict partial order on $X$ if $R$ is irreflexive, antisymmetric, and transitive.

We will now describe another type of relation known as an equivalence relation.

Definition: An Equivalence Relation on a set $X$ is a relation $R \subseteq X \times X$ that is reflexive, symmetric, and transitive.

If $X$ is a set and $R$ is an equivalence relation of elements in $X$ then we sometimes use the notation "$\sim$" instead of "$R$".

Perhaps the simplest equivalence relation can be defined on the set of real numbers $\mathbb{R}$. Define the equivalence relation $=$ such that for all $x, y \in \mathbb{R}$ we have that the pair $(x, y)$ is in the relation if the numerical value of $x$ is the same as the numerical value of $y$.

Clearly $=$ is reflexive since for all $x \in \mathbb{R}$ we have that $x = x$. Also, $=$ is symmetric since if $x = y$ we have $y = x$. Lastly, $=$ is also transitive since for all $x, y, z \in \mathbb{R}$ we have that if $x = y$ and $y = z$ then $x = z$. Therefore $=$ is an equivalence relation on $\mathbb{R}$.